The natural logarithm is the logarithm to the base e, where e is approximately equal to 2.71828... (no precise decimal fraction can be given, as e is an irrational number). The natural logarithm is defined for all positive real numbers x and can also be defined for non-zero complex numbers as will be explained below. Although this function was not introduced by Napier, it is sometimes known as the Naperian Logarithm.
Mathematicians generally understand either "ln(x)" or "log(x)" to mean loge(x), i.e., the natural logarithm of x, and write "log10(x)" if the base-10 logarithm of x is intended. Engineers, biologists, and some others write only "ln(x)" or (occasionally) "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x).
Most of the reason for thinking about base-10 logarithms became obsolete shortly after about 1970 when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(x)" to mean the base-10 logarithm of x and use only "ln(x)" to refer to the natural logarithm of x. As recently as 1984, Paul Halmos in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) At the time of this writing (2003), many mathematicians have adopted the "ln" notation, but "log" also remains in widespread use.
To avoid all confusion, Glasglow uses the notation ln(x) for the natural logarithm of x and log10(x) for the base-10 logarithm of x.
Logarithms can be defined to any base, not just e, and they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.
Initially, it seems that the base-10 would be more "natural" than base e. The reason we call ln(x) "natural" is twofold: first, the natural logarithm can be defined quite easily using a simple integral or Taylor series as will be explained below; this is not true of other logarithms. Second, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow to describe growth and decay behaviors), and so the natural logarithm is more useful in practice. To put it concretely, consider the problem of differentiating a logarithmic function:
Formally, ln(a) may be defined as the area under the graph (integral) of 1/x from 1 to a, that is,
This defines a logarithm because it satisfies the fundamental property of a logarithm:
The number e can then defined as the unique real number with ln(e) = 1.
Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, meaning ln(x) is that number for which eln(x) = x. Since the range of the exponential function is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive x.
The derivative of the natural logarithm is given by
One may define ln(z) also for all non-zero complex numbers z. The above Taylor expansion remains valid for all complex numbers x with absolute value less than 1. If the non-zero complex number z is expressed in polar coordinates as z = r eiφ with r > 0 and -π < φ ≤ +π, then
A somewhat more natural definition of ln(z) interprets it as a multi-valued function: for z = r eiφ we set
The preferred way to deal with multivalued functions like this in complex analysis is via Riemann surfaces: the function ln is then non defined on the complex plane but instead on a suitable Riemann surface having countably many "leaves" and the values of the function differ by 2πi from leaf to leaf.
The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact:
Here is an example in the case of g(x) = tan(x):
where C is an arbitrary constant of integration.